You need to change your main account status to public, and then set individual albums to private. Here are more details from the Photobucket help:
You can edit your album privacy settings on the All Albums page. Your main account status must be set to Public before you can change the privacy setting of individual albums. See Change your account privacy settings for instructions. If your account is public, you can change the status of individual albums, at the first level, within your account.
Log into your Photobucket account.
Click All Albums in the secondary
menu. A list of all your albums
displays. Mouse over the album whose
privacy you want to edit. Click the
edit link. The Edit Album pop-up
displays. Check it out Choose whether
the album is public or private. Notes:
Only albums at the first level can be
set to public or private. Albums at
the second, third, etc. level inherit
the privacy setting of the first-level
album within which they exist. If you
make an album private, you can add a
guest password. To test guest access,
you must log out of your Photobucket
account. Make sure any guest passwords
you assign are different than your
main administrative password. If you
make them the same or similar, your
chances of logging into your account
as a guest or someone guessing your
master password are more likely. When
you assign this password, use a
combination of letters and numbers for
added security. Passwords are case
sensitive, so a combination of upper-
and lower-case letters is also good.
Click Save. If you do not click Save,
your changes are discarded.